When
3:30 – 4:30 p.m., Jan. 12, 2026
Where
Many models of collective behavior in economics, engineering, and the applied sciences involve large populations of agents whose dynamics are driven by individual optimization and strategic interactions. Mean field game theory provides a rigorous framework for studying such systems in the limit of infinitely many agents, leading to coupled Hamilton–Jacobi and continuity equations that describe equilibrium behavior at the macroscopic level.
In this talk, I will present recent advances in the analysis of mean field games and related differential games in the presence of state constraints, a feature that is intrinsic to many realistic applications, including bounded spatial domains, feasibility constraints, and resource limitations. While unconstrained mean field games are by now well understood, state constraints introduce substantial analytical challenges, such as loss of regularity, boundary singularities, and difficulties in interpreting the transport equation governing the population evolution.
I will describe a Lagrangian approach to constrained mean field games, in which equilibria are formulated in terms of probability measures on admissible trajectories. This perspective allows one to establish existence, uniqueness, and regularity results and provides a rigorous interpretation of the underlying PDE system even in nonsmooth settings.
I will then introduce a complementary research direction focused on competitive interactions at the level of evolving distributions, including zero-sum differential games in which the state variable is a population density rather than a finite-dimensional state. These models give rise to infinite-dimensional Hamilton–Jacobi–Isaacs equations and suggest new equilibrium concepts for competitive multi-population systems. I will conclude by outlining ongoing and future directions, including multi-population mean field games, population-dependent and time-varying constraints, and extensions to nonlinear and nonholonomic dynamics.